Homogeneous verbal functions on groups

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

B ENEDETTO S CIMEMI

Homogeneous verbal functions on groups

Rendiconti del Seminario Matematico della Università di Padova, tome 49 (1973), p. 299-321

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BENEDETTO SCIMEMI

(*)

1. - This research was

originated by

^our

previous

paper

[4],

^whose

main results will be

shortly

^{recalled: let p be a}

prime number,

^{P a}

finite p-group of

exponent p-, d

^{a divisor of}

p -1, a

^a

primitive

d-th root of

unity,

^{mod an}

automorphism

^{of P of order}

d;

^then

every element h E P ^can be

uniquely

^{written as}

The set

P1=

need not be a

subgroup

^{of P}

(we

^{could take}

any other

P; .

⁼

(h, ;

^{to avoid}

trivialities),

^{but a}

loop operation

^{o can be} introduced on

Pl by letting

^{for all} x, y ^E

P,,:

x o y ⁼ If this element is

expanded

^{in terms} of basic commutators

in x, y

then a recursive formula

naturally yields

^{for the}

egponents eu

^some

rational

(non-integer) values;

^all

examples

^{show evidence that these}

numbers are

independent

of x, y,

P, q

^and

only depend

^{on d.}

The motivation for the

present

^{research was to} prove this inde-

pendance,

^thus

showing

^{a more natural}

origin

^{for the}

loop

^structures

we studied in

[4]. After

^we noticed that we could assume without loss ~’ =

(Pl),

^{it seemed natural to look for a} group F**

admitting

(*) ^{Indirizzo dell’A.: Seminario} Matematico dell’Universith - Via Bel.

zoni 3, 35100 Padova.

Lavoro eseguito nell’ambito dei

Gruppi

di Ricerca Matematica del C.N.R.

an

automorphism ø

such that: if P is any finite p-group with n gen-

erators, and 99

^{is an}

automorphism

^{of P}

inducing

the a-th power ^on these

generators,

^{then P is a}

homomorphic image

^{of F**}

and 99

^is

induced

by

^0.

By using

^{a recent} theorem of R. H. Bruck

[1],

^a

satisfactory

^answer

was found. The main theorem of the

present

paper

(theor. 1,

par.

5)

states the

following:

Let F be a free group of finite

rank n, generated by

^the free gen- erators

X, Y,

^{.... For each}

positive

number d there exist groups F*

and .F’** with the

following properties:

1) .F~.F*~h**.

2)

The elements of F* may be

uniquely represented

^{as infinite}

ordered

products

^{in terms} of basic commutators in

X, Y, ...

where the

exponents e.

lie in the

ring Z[d]

of those rational numbers

rls

^{such that no}

^prime dividing s

is =1 mod d.

3)

The element of ~** are

similar, with e.

^{in the}

ring

obtained from

by adjunction

^{of a}

primitive (complex)

^{d-th root}

of

unity

^{a .}

4)

In 1** the ^«

a-th-power » g"

^{is defined}

(for each g

^E

F**)

^and

there exists an

automorphism 0

of ~’** such that

5) Every

^element

g E F* (in particular,

every word in

~Y, Y, ...)

can be

uniquely

^{written as}

Now let p be ^a

prime number,

^{P a} finite p-group of

exponent

pm with n

generators x, y,

^{..., d a} divisor of a

primitive

^{d-th root}

of

unity,

^mod pm. Then there exists a

unique homomorphism

^{v of F**}

onto P such that

If q

^{is an}

automorphism

of P such that X9, ⁼ xa,

ygl

⁼ ya, ^... it is clear

that 0153fP == xa =

(Xv)a

⁼

(Xa)v

⁼

(X~~v,

^{etc. Hence it is}

easily

^{seen that}

for _every = gv ^{we as we wanted.}

The consequences of

5)

^{are derived in terms of} « verbal functions ».

A verbal function on a group ^is the

analog

^{of a}

polynomial

^function

on a

ring.

^{Let H be a} group, I’ the free group with free

generators Xl, X2, ..., Xn;

^{to each word}

f = f (X1, X2, ..., Xn) E .~’

^{we associate}

the « verbal function in n variables

say 7, mapping

^the

n-ple ( h1, h2 ,

^{... ,}

hn )

of elements into the element

h2 , ... , hn ) E H

which is obtained from

f by replacing

^each X

by hi

^and

taking

^inverses

and

products

^{in H. For n ==}

1,

^a verbal function is

simply

^a power;

for n ⁼ 2 among the verbal functions

(or

^verbal

operations)

^{we find}

the usual

group-multiplication

^and

conjugation,

associated to the words

Xl X2

^{and I etc.}

Clearly,

^the verbal functions in n variables on H form a

group -P (with respect

^to

multiplication

^of

images),

an

image

of F under the

homomorphism

^{If H is a finite}

p-group,

p ---1 mod d,

^{then there is a} natural way ^to extend the

homomorphism f - f

^to the group P**. If ^we

apply

^this

homomorphism

to the word _g, factorized as in

5) above,

^we derive the

following (Cor. 1,

par.

7):

Let P be a finite p-group of exponent

pm, d

^a divisor of

p -1,

a a

primitive

^{d-th root of}

unity, mod pm.

Then every verbal func-

tion g

^{on P can be}

uniquely

^{written as with}

By analogy

^with

Analysis,

^a verbal function

satisfying

^{the above}

condition

for g;

^{is said to be «}

a-homogeneous

^of

degree j

^{». Since}

(but

ⁱⁿ

general g; ~ I’),

^the

homogeneous factor g;

is obtained from a basic commutator

expansion

^{which is}

uniquely

^defined

by

the word g and the

number d, through

^a substitution of elements of P.

If we

apply

^{this to the} group

multiplication

^on

P,

^thus

letting

we find for

gl

^a

loop-operation

^{on P}

(Theor. 2,

par. 7). ^In

particular,

^{for d =}

2,

^we

find,

^within

isomorphisms,

^the

loop operation

^{which has been}

extensively

^studied

by

G. Glauberman

(see

^{ref. in}

[4]).

^{When P admits an}

automorphism

^{of order}

d,

^this

operation

^is

easily proved

^{to induce on the «}

eigensubset &#x3E;&#x3E; Pi

^{the same}

multiplication

^{we defined}

above,

^i.e.

Apart

^from

loops,

^{the whole}

subject

^{in the}

present

^{broader context}

is

strictly

connected with the work of M. Lazard

[3]

^on

nilpotent

groups and

Lie-algebras;

the relation between his « suites

typiques »

and our

homogeneous

verbal functions is

explained

in par. 8:

roughly speaking,

^some of his results can be obtained from ours

by letting d

tend to

infinity.

2. - In this

paragraph

^we consider the « binomial &#x3E;&#x3E;

properties

^of

some

rings;

^{some of the}

following elementary

remarks may be well-

known,

^{but we do not know a}

complete

^{source to refer to.}

Let I~ be an

integral

domain of characteristic zero. We shall denote

by

^its

ring

of fractions with

integer

denominators. If the element

as follows:

is defined

DEFINITION. A ^« binomial

ring »

^{is an}

integral

domain C of char- acteristic zero, such that is in C for every ^c in C and every ^non-

negative integer

^k.

Of

course, Z

^{is a binomial}

ring (the

^formula

accounts for

negatives) . However,

^{if x is an indetermi-}

nate over

Z,

^{then the}

polynomial ring Z[z]

^{is not} binomial. Therefore

we introduce the set

(Polyals « ganzwertige Polynomen &#x3E;&#x3E;) .

It is well-known that consists

precisely

^{of those}

polynomials

^{in which assume}

integer

values whenever x is

replaced by

^an

integer.

This characterization has the immediate consequence is a binomial

ring.

^More

generally,

^{if Z is}

replaced by

any binomial

ring C,

^{we define}

and

easily

check that

C f x}

^{is a binomial}

ring. Clearly,

^{if u is an element}

of a

ring containing C,

^then

C f ~} _ C{x}}

^{is a minimal}

binomial

ring containing

both C and u

(thus

^{is the « binomial}

closure &#x3E;&#x3E; of

C[x]).

^{If x2 , ... , xn are}

algebraically independent

^elements

over

C,

^{it is natural to define}

by

^induction

An

element e

^{of this}

ring

^{can be}

expressed

^{as a} linear combination

I I , I I

with coefficients in C of

products I

^{where the}

^kils

^are

non-negative integers ;

e is

characterized by the properties : ... , x~ ]

i

... , cn) E C

for all

c,eC. In [1 ],

^an element of

7~~x~ , x2 , ... , x~,~

is called an

integer-producing polynomial

^{». In}

particular,

^since

2/}

⁼

_ ~ f x~ {y~,

^{we must have « binomial} identities » as

for some

LEMMA 1.

Any subring (with 1) o f

^binomial.

PROOF. Such a

ring

consists of all the fractions

rls

such that the

prime

^{factors of s}

belong

^{to a fixed set of}

primes. Then, by

^{the binomial}

identities

above,

^{it suffices to} prove that for any

prime p

^{we have}

1/p

^the

^subring generated by 1/p.

^{To this}

aim

^{we write}

k ! ⁼ p-r ^with

p ~

r ; then

1 /p

^{= a + rc for some a e}

Z,

^{c e}

hence, by

^{the binomial}

expansion

^for

x + B

^k

y ,

^{/ it suffices to prove ? /}

when In fact we

compute

If d denotes a

positive integer,

^we shall denote

by Zed]

^the

ring

of the rational numbers

r/s

^{such that no}

prime dividing

^{s is =1 mod d}

(e.g. ^Z[2]

⁼

^{~~21 ~ Z[4]}

⁼

1, t, ...]).

LEMMA 2. Zet oc be a

primitive (complex)

^{d-th root}

o f unity. For

every

non-negative integer k

there exists a

polynomial e,

^{e such}

i

that k = _k ^{f or}

^{all i c- 7 .}

PROOF. Let us write k ! ⁼

ts,

^{where the}

prime

factors of t and s

are =1 and

respectively #

^{1 mod} d. In the factor

ring

^{the d-th}

roots of

unity

^are

exactly d,

say where is a unit if mod d

(for

^{t = p-} this is in Lemma 1 of

[4];

then the Chinese remainder theorem

applies).

^{In let us}

divide and write

where Î’k

^E

Z[xl

^{is a}

polynomial

^of

degree

smaller than c.

By

^substitut-

ing x

^{= ai we obtain}

Since we deduce

From this we shall deduce yk ^E and

as we wanted. In

fact,

^{let us denote}

by y,,

^the

image

of y,, under the natural

homomorphism

^{of onto We know that}

for

i =1, 2, ... , d ;

^therefore

yk

^{is a}

polynomial having

d distinct roots in and

degree

smaller than

d ;

moreover, the difference of any two such roots is a unit in

Z/tZ,

^thus

yk = 0

^{i. e. yk E}

LEMMA 3. ⁼

PROOF.

By

^Lemma

1,

^is

binomial; by

^Lemma

2,

^contains

(X

^B

k

^{for every k. Since}

^1/d

^lies

^inz[dl

^{I we As}

for the inverse

inclusion,

^{we must}

only

prove or,

equivalently,

whenever p is ^a

prime, p - 1

^{mod d.}

Since this is trivial when p ⁼

d,

^{we can assume p}

= dq --~- h,

^{1 C h d.}

Since all the elements of the field are roots of x, ^we have

Then ⁼

-1 ) xh

^we derive that the remainder

/ uB

of the division

of I

Replacing x by

^aa

yields

We now sum over i and take into account that ^I whenever 1. Then Lemma 2

yields

As a binomial

ring containing

oc, contains the left hand

term,

thus also

lip,

^since

p ,~

^d.

Let us notice that ai ^- mi is a unit in

whenever I fl j

^{mod d.}

This is

easily proved by letting r

^{= i}

- j

^{in the}

following

^relation

LEMMA 4. Let

p be a prime integer, p ---1

mod d..Let m be any naturale

number, a a primitive

^{d-th root}

of

^{1 mod} pm. Then there is a

unique ring homomorphism of

^onto

mapping

^{Lx into}

PROOF.

Any

element of Z[,] is written as a fraction

rls,

^with

p ,~ s ;

therefore s is a unit

mod pm

and hence the natural

homomorphism

of Z onto

Z/p-Z

^is

uniquely

^{extended to the}

ring

of fractions Since is

isomorphic

^to the factor

ring

^where

^ød

^is

the

cyclotomic polynomial,

the statement will hold if ^we prove

~d(a) ---

⁰

mod pm.

From xd -1

= TI ød,

^we derive that

Ød(a) =t=

⁰

, d’jd

modpm implies ~~. (a) = 0 modpm’

^{for some}

Since divides

xd’-1,

^this

implies ad’

which is im-

possible,

^{as a is a}

primitive

d-th root of 1

mod pm’ (see ^[4],

^Lemma

1).

Thus ^- 0

mod pm.

3. - ^We shall

operate

^{within a} group whose existence _{and pro-}

perties

^{are the}

subject

^{of a} recent paper

by

R. H. Bruck

[1];

^{from it}

we borrow definitions and

symbols,

^{and refer to it for details. Here}

is the statement of Theor. 1.1 of

[1],

^{in a}

slightly simplified

^form:

Let

(A, )

^{be a}

simply

^ordered

set, consisting

^{of a} finite number n &#x3E; 0 of

elements; (A~, )

^a universe of basic

symbols,

^{based on}

A;

Am

the subset

consisting

^{of the elements of}

length L(u)

^{= m.}

Let

(.F, ~ )

^{be a} free group of rank n ^{on a} free set of

generators

the

corresponding

^{set of basic}

commutators;

the subset of commutators whose total

weight equals

^m.

If C denotes a binomial

ring,

then there exists a group FO whose elements may be

uniquely represented

^{as ordered infinite}

products

, then for each we have

where ... uk ^are the elements of

A~

^of

length

smaller then

L(u)

^and Pu ^{is a}

uniquely

^{defined element of ... ,} xk; ^I i.e. an

integer-producing polynomial.

Fe is an extension of

.F’,

which is in fact

represented by

^elements

11 go-

^{with all}

exponents eu

^{in Z}

(as

ⁱⁿ

^[2],

^theor.

11.2.4).

In Fe the «

c-th-power

^» gl exists for every

g E F°,

^{c E}

C;

^if

The fact that

Pu

^and Ru ^{do not}

dep end

upon C

implies

^{that for}

their determination one can assume C = Z and

operate

within the free

nilpotent

group

(here .Fm

denotes the m-th term of the lower central series of

.F’) by

^the

«collecting

process »,

Let us

point

^{out some} conseguences of this theorem

(not

^all

explicitly

^{stated in}

[1]~ :

The c-th power

enjoys

the basic

properties

^of

integer

powers, ^i.e.

the last formula

being

^a

generalization

of P. Hall’s

([2], 12.3.1).

By analogy

^{with the}

language

^of

C-modules,

^{one can introduce}

the

concepts

^of

C-groups (i.e.

groups with c-th power for ^{c E}

C), C-subgroups, C-homomorphisms

^etc.

by requiring

^the

compatibility

with

taking

c-th powers. If ^one does so, ^some basic

properties

^{of F}

are

naturally generalized

^to

FC;

^for

example:

the lower central series of Fc consists of

C-subgroups,

^{each term}

F§’ consisting

^{of those}

elements

gl--

for which eu = 0 ^when the

factor-group

is a free

C-module;

^since

nF’

⁼

(1),

^{a statement}

holding

m

on may be

proved

^{to hold on}

FC, by

^{induction on m. These}

and others similar statements can be

proved by

^{a common}

procedure,

as follows: a

property

of F is r educed to a set of relations among the

(integer) exponents

^{involved in the}

infinite-product representation.

Because of their

polynomial

nature, ^{the same} relation must hold when Z is

replaced by C,

^thus

yielding

^the

analogous property

^{for Fe.}

In par. ^{4 we} shall exhibit some

examples

^{of this}

procedure.

Fc is called a «

completion

^{of F » with}

respect

^{to the}

ring

^{C. Here}

«

completeness »

^means

substantially

^{that if}

hl, h2,

... ,

h~ , ...

^{is a} sequence of elements of Fc such that for

every

then the sequence of

partial products

converges ^{to a}

unique

^{element in other}

words k == kj modFf

for

every j.

^{Therefore some infinite}

products

^are

meaningful

ⁱⁿ

FO;

for

example,

^{the ordered}

product

^{is a} well-defined element of

F, ^provided

^the

following

condition holds: if

L(u)

⁼ m, then

Fc.

^In

fact,

the finiteness of A

implies

^that

only

^a finite number of basic commutators have

weight

^m.

4. - One of the basic

properties

of the free group F

implies

^that

for any

integer

^c there exists ^a

unique endomorphism

^{of F}

mapping

every ^free

generator ga (a

^E

A)

into its c-th power,

The purpose of this

paragraph

^{is to} extend in the natural way this

endomorphism

^to

h~, letting

^{c assume} any value in C. To this

aim,

we start with the basic commutators and

define,

for any

c E C, v

^E

A.

LEMMA 5:

PROOF. Since

(c~)

^{is true}

by

definition if

L(v)

⁼

1,

^we shall induce

on m ⁼

.L(v) .

^Then we can assume

Therefore ^we calculate

(compare

par.

3)

where

where

Similarly

^we calculate the commutator and prove statement

( a ) .

b )

^{It is a} consequence of well-known commutator-relations

([2],

^{18.4.10 and}

foll. )

that in the

group F

^{we have ---}

g:m

whenever

m = L(v).

^{This is}

equivalent

^to

8v,v(c)

⁼ cm, ^{= 0} for

L(u) L(v)

^or

Z(~) = Z(v), ~ ~ v

for any ^c

integer.

^is

a

polynomial,

^{the same must hold when c assumes} any value ⁱⁿ the

ring

^C.

q.e.d.

Now let be any element of FC. We define

First of

all,

^{we must} prove that there is ^no

problem

involved with the infinite number of factors.

According

^to the final remarks in par.

3,

^{it will suffice to show that}

(gl,")e-

^E

FO

^{if m =}

^L(v).

^In

^fact

we claim

This follows from Lemma

5,

^{if we set}

LEMMA 6. Let

ments

of Aoo

^whose

length

is smaller than m, and

Qu E

x1, ... ,

PROOF. In order to calculate

fu

^we

only

^{have to} consider the finite

product

^{v. Since we have}

just

^{seen that I}

a

repeated

^{use of the}

product

^formula

(involving

^the

polynomial Pu)

combined with Lemma 5

yields (b).

LEMMA 7. Let c E C. Then the

mapping g

^r~

C-endomorphism o f

^FO.

I f

^then

PROOF. Let )e elements of po. We write

and use Lemma 6 to show that

where

But

when 9

^{E F and c E}

Z, by construction,

^the

mapping g

^r-~

gici

induces an

endomorphism

^{of F. Therefore s,, = t~ in} this case, hence also

T u by

^{the familiar}

argument,

^and

(gg’)to

⁼

The

proofs

^that

(gccl)e

^{e - and =}

g(cel

^are

similar,

^involv-

ing polynomials

in c, e, eUl’ ^{... ,}

We conclude this

paragraph

^{with a} definition and some remarks which will

simplify

^the

proof

^{of our} main theorem in par. 5.

DEFINITION. A

mapping

^{E :} will be called a

C(z)-mapping

if for each u E

A.

there exists an

element Eu

^{E such that}

for every c E C .

For

example, if g

^{is a} fixed element in

FIO,

^{then the}

«exponential

function » c - gC ^{is a}

by

^Lemma

6,

^{the same holds for}

More

generally,

^our

previous

remarks show that

products

and powers of

C{x}-mappings

^are still such.

5. - Let c be an element of the binomial

ring C, g -

^{the endo-}

morphism

of Fc of Lemma

7, j

^a natural number. The element will be defined to be «

c-homogeneous

^of

degree j

^{» if}

gc’

(i.e.

go is ^{an «}

eigen-element » belonging

^{to the}

« eigenvalue » Ci).

^The

term is

suggested by

^the fact that such an element will

produce

« c-homogeneous

^{functions » in the sense} of par. ^1.

Let d be a

positive number,

^{and the}

rings

defined in par.

2;

^we

apply

the construction of _{par. 3} with these

rings

^{in the}

role of C

(by

^{Lemma 1 and 4 both are}

binomial)

and consider the groups Then F is

naturally

^{embedded in}

FZ[dJ

^{and the}

latter in

The main result of this paper is

THEOREM 1.

Every

element g ^E

Fz Idi

^{can be}

uniquely

^{written as}

i.e. a

product o f a-homogeneous

^elements

of degree j

⁼

1, 2,

... , d.

PROOF. Let us write for short ⁼ 1~’** ⁼ It may be worth to

point

^out

that gi

is to be found in I’*

although ga"’

^is

only

^{defined in F**.}

By

^Lemma

7,

^a

a-homogeneous

element of

degree j

^{is also}

ai-homogeneous

^of

degree j

^{for all}

integers

^i.

We shall

operate mod-F-**,

and induce on m.

For ^{m ==} 1 we define:

It follows from Lemma 5

(or directly

from the definition

of g

that any element of .F’** is

a-homogeneous

^of

degree 1, mod F’:*.

Therefore we have

the last congruence

being

^trivial

for j .--- 2, ... , d.

The elements gl,;

are in J~* and

they

^are

unique

^since

for j = ~, ... , d

^the

condition

(gl,~)" = gi:~ --- (gl.~)"’ ^implies

^thus

- 1

Then the other condition

implies

also g1,1 ^-= g.

Let us assume that we have

proved

the existence of d elements which are

unique mod 0f/*

^with

respect

^{to the}

following

conditions

We must construct d elements E F*

satisfying

similar congruences and prove their

unicity

^mod

I’m+1.

^{To this}

aim,

^{we shall}

compute

the «mean value » of for c

running

^{over the d-th roots of}

unity.

^{Thus let us define}

We now consider the

mapping

^of into F** defined

by

Since lies in

F*,

the final remarks in par. ⁴ show that is

a i.e.

By letting

^{c = (Xi}

(i = 1, 2, ... , d),

the inductive

assumption implies

~m,~(ai) ---1

^therefore

8,,(ai)

^{- 0 if}

L(u)

^{m. Thus we have}

If we take into account that these ele- ments are central we can

compute:

We want to prove that the

exponent of g,~

^{lies in}

In

fact, by

^{Lemma 2 for each}

positive integer k

there exists a

ai

polynomial

^{such it follows that for}

every there is a

polynomial

such that Since for

d

I as we wanted. Therefore ^we have

proved For j =t= m mod d

^{let us define from}

by simply cancelling

all factors of

weight greater

^{than m in its infinite-}

product representation; for y ==

^{m mod d let us define}

yve must now prove the congruences We have

by

^definition

Hence

by

^{Lemma 7 we}

compute

Here we have used the fact that for

all j

^{we have}

so that

(mod .~’m+1)

the order of the factors in the

product

^is

irrelevant,

^{and the}

exponent

^(Xj may be taken ^out of the

parenthesis.

We still have to consider the ^m mod d. First we notice that for m the construction above has

given

⁼

then the inductive

assumption

g --- gm-1,1... gm-1~d combined with the definition of gm.3

yields

^{also Since we}

have

proved

^the congruence ^we deduce that we can write

where z,,,,

^{is a}

product

of powers of basic

commutators g.

^of

weight L(u) =

^~n. From Lemma 5 we know that for such commutators we

have

This

gives immediately

^and

finally

As for

unicity,

^{let us assume that}

satisfy

the same

requirements

^{for -.}

By

the inductive

assumption

^{we can}

write,

Since the

previous argument gives

^{we can}

compute

^{in two} different ways,

namely:

By comparison

^we

get ~2U~.,,,~~ai-"~ -1,

^hence

- 1 whenever j K m mod d.

^As

for 1 == m,

^{the first condition} g -

hm,1 ... hm,d implies wm,l . ·.’Zl~m,d ---1,

^thus

~,vm, f =1.

Thus the theorem is

completely proved.

Let us remark that

along

^the

proof

the power

mappings

g ^-

g",

have both been used.

By

^Lemma

3,

^{is a minimal}

binomial

ring containing

^{a and}

Therefore,

^{if this}

proof

^{has to}

be carried on in a group of

type

^{we cannot choose for C a smaller}

ring

^{than For the} same reason we cannot

expect

^{to find the}

homogeneous

factors gj in ^a proper

subgroup

^{of F* =}

FZcdH

^even

if g lies in F.

6. The

proof

of theorem 1

gives

^a recursive method to

compute

gm,;

starting

^{with so} that gj ^can be determined mod

2~

^for

arbitrary

^{m. We shall}

apply

^{the first}

steps

^{of this}

procedure

^{to the}

element X Y of the free group F ⁼

X, Y~ ,

whose elements are re-

presented by

ordered infinite

products

^as

We first choose d = 2

(d = 1

^is

uninteresting).

^{Then a = -1 and}

=

Zr2] = Z[’2-1-

^We

compute :

Let us now choose d ⁼ 3. We

compute

For d ⁼

2, 3,

^{4 and within}

weight

⁴

(i.e.

^mod

.F’b )

^{we find the same}

exponents

appearing

ⁱⁿ

[4],

p. 215. The ^reason for this coincidence

(the computing

^{method was there}

fairly different)

^{will be}

explained

in par. ^7. From theor. ³ it will also be clear

why

^{two different values}

d d’

yield

^{the same elements for}

The case d ⁼ 2 is

special:

^{if we start from a}

word g

^{E F we can}

avoid recursive formulas and basic commutators

expansions,

^{since the}

« mean value » of over the two roots ~1

gives directly

^{the exact}

formula for g1. In fact let ^us write

We claim:

f 2 = g2 .

^We

easily compute:

By unicity

^{we must have}

f 1=

911

f

⁼ g2 ^{as we} wanted.

In

particular, if g

^{= X Y we find}

7. We shall

apply

^{theor. 1 to} finite p-groups;

possible general-

izations will be discussed later. In this

paragraph

^{P will}

always

^denote

a finite p-group of

exponent pm, d

^a divisor of

p -1, a

^a

primitive

d-th root of

unity,

^y

mod pm.

There is a

unique

way ^to

assign

^{to P a} structure of

i.e. for any

h E P,

^{we define he to be the}

unique

^element

1~ E P such that ks ⁼ hr.

Equivalently,

^we may consider the canonical

ring-homomorphism

^{of Z onto}

Z/p-Z,

^{extend it to the}

^ring

of fractions

Zed] by letting

^{c =}

rls r-~ (r

+ + ^{= b} +

pmZ

and define he ⁼ hb. P becomes also

(although

^not

canonically)

^{a if}

we define h" ⁼ ha i.e. we use the

homomorphism

^{of Lemma 4.}

We now choose n

generators

zi , x2 , ... , xn for P and denote

by F

the free group with free

generators

g1, g~, ... , gn . Then ^{we see} that

there is ^a

unique

^{of _ .F’**N}

onto

P

such that

° ’

If v is such a

homomorphism,

^it is clear how it must act on a basic commutator _gu; as v is also then we must have for any

where

only

^{finite factors are non}

trivial,

^{as P is}

nilpotent.

^This

accounts for

unicity.

^{As for}

existence,

^we may

define gv by

^{the last}

equation above;

^the

proof

^{that v is a}

Z[,,][a]-homomorphism

^follows

easily,

^{y once the}

egponents eu

^{have been}

replaced by

^proper

integers,

i.e.

by

^the

ring homomorphism

^{of on we} described in Lemma 4.

As we have

already

^{seen in par.}

1,

^this

implies

ⁱⁿ

particular

^that

if (p

^{is an}

automorphism

^{of P =} z~ , ... ,

Xn)

^{such that =} xt" for

i = 1, 2, ..., d then 99

^{is induced on}

by

^{the automor-}

phism 0:

9 F*

9 (,x)

^{of .F’**.}

Now we reconsider the

homomorphism g 1--+ g

described in par.

1, mapping

the free group I’ ^onto the group

P

of the verbal functions in n variables on the p-group P. As

before,

there is the

possibility

to extend this

mapping

^{to a}

Z[d][a]-homomorphism

^{of F**}

^(the

^finite

p-group P

plays

^now the role of P

above):

^if

for any

n-ple (hl,h2, ... , hn )

of elements

h i E P

^{we define}

Clearly,

^if

gjllll

⁼

g"’

ⁱⁿ

^.F’**, then ij

^is

a-homogeneous

^of

degree j,

^i.e.

Then from Theor. 1 it follows

COROLLARY 1. Let P be a

f inite

p-group

of

exponent p-, ^{d a divisor}

of p -1, a a primitive

^{d-th root}

of

¹

mod pm.

Then every verbal

f unc-

tion

(in n variables)

^{on P can be}

uniquely

^{written as a}

product of

^d

a-homogeneo2cs

^verbal

functions of degree 1, 2,

^{... , d.}

PROOF.

Let 9

^{be a} verbal function on

P, corresponding

^{to the}

word We embed .F in .F’** and write g = gl g2 ... ga, ^{I as} in Theor 1. Then gj may be ^not in

.I’, but gi

^{is a} verbal function on P

(i.e.

^{there is a}

word 9, /

^E

^.F’,

⁹

although depending

^on

P,

^{such that}

91 = g3)

^and

^clearly being a-homogeneous

^of

degree 1.

As for

unicity,

^let us assume also

liEF, ij ^a-homoge-

neous of

degree j.

^We shall induce on the class c of P

(or P),

^thus

assuming 1i == gi mod Pc .

^{Then if we set the element}

..., h~)

^{lies in the center} of .P for any hence:

Moreover,

^the

assumption ~1 ... ~a = gl ... 9a ^implies 21(h,

^...,

hn)

^...

...

2,,(h,

^{7 ...,}

^hn)

^{= 1. If we}

^replace

^each

^{hj by}

its power

h11

^we

^get

for the

elements k, = z ~ ( hx , ... , hn )

^the

following

^relations:

These are

equivalent

^{to a set of d}

homogeneous

^linear

equations

ⁱⁿ

the

ZlpmZ-module P,,,

whose coefficients have the

(Vandermonde)

determinant Since ai - ai is a unit

mod pm,

^{there is}

«

only

^{the trivial solutions} 1 ⁼

k2

= ... = kd ^{= 1. Therefore = 1 and}

;j

^{as we wanted.}

When the

word-operation g

is the usual group

product,

^associated

to the element

9 = XY

of the free group .F =

(X, Y~ ,

^{then Cor. 1}

applies

to the situation we studied in

[4],

^and

unicity

^forces

j,

^{to be}

the same

operation

^{which was} there denoted

by

the circle _o; in

fact,

if 99

^{is an}

automorphism

^{of P and}

x, y E P ;

^then

yields

where the fact

that ii

^{is a verbal} function has been used to write the first

equality.

Therefore the definition in

[4]

^was:

x o y = gl(x, y).

This settles a

question

^{which arose in}

[4],

in connection with the

independence (upon P, a etc.)

of the rational numbers listed in the table of

[4],

p.

215; actually

^these

exponents

^{must be}

independent,

as

they

^are

uniquely

^defined

by

the factorization

of Theor.

1,

and this is

uniquely

determined

by

^the choice of d.

Since this factorization is

possible

^{in but not in}

F,

^{it is also}

clear

why

^the

independence

^{of the formulas}

requires

^{the use of rational}

non-integer exponents.

Once this coincidence has been

recognized,

many

arguments

^of

[4]

may be

repeated

^without substantial

changes ;

ⁱⁿ

fact,

^{the automor-}

phism

(p is ^no

longer available,

^{but the first}

steps

^{of the}

proof

ⁱⁿ

Theor. 1

imply

^{x 0} y --- xy mod

x, Y)2’

^{hence x o z =} xz, ^{x o}

(yz)

⁼

(0153 0 y)

^z

whenever z commutes with x, y ; ^on these relations most of the

proofs

in

[4]

^are based. The result is the

following.

THEOREM 2. Let P be ^a

f inite

p-group.

If

^{we choose a divisor d}

of p -1,

^write

XY==glg2...gà

^as in T heor. 1 and

define

x o y =

=

91(X, y) for

each x,

y E P,

^{then we obtain a}

loop P, 0

^{with the}

following properties:

a) P,

^{0 is}

power- associative;

the order

of

^an element is the same as in the group.

b) If

^the

3-generators subgroups of

^{P have}

nilpotency

^{class c e}

then

P,

^{0 is}

centrally nilpotent of

^class

[(c

+

d - 1)[d] ([...] = integer part of ...~ .

c) I f

^{a is a d-th root}

of

¹

mod pm,

^the

exponent of P,

^{then the}

power

mapping:

^{is a}

loop -automorphism of P,

^o.

Here are some

particular

^{cases: if d =}

2,

^{we find x 0} y ⁼ the

loop

of Glauberman

(see

^{ref. in}

[4]) ;

^{if d = p - 1 and c C p,}

then

P,

^{0 is an} abelian group

(as

^we shall see, this is the additive _group

of the

Lie-algebra

associated to

P, according

^{to Lazard}

[3]).

^Since

x - y ^E

x, y~,

any

subgroup

^{of P is a}

subloop

^of

P, o ,but

^{the converse}

is not

true;

^for

example,

^{the set}

^Pi

^{of the «}

eigenelements

^{» for an}

automorphism

of order d

(see

par.

1 )

^{is a}

subloop

^{but need not be}

a

subgroup.

Since x o y is ^a verbal

operation

^on

P,

^a

group-auto- morphism

^{of P is a}

loop-automorphism

^of

P, o; again,

^{the converse}

is not

true,

^the

mapping being

^a

counterexample.

^The

following application

may deserve ^some attention: if the

3-generators subgroups

of the p-group P have class p, ^then the group of the

automorphisms

of P is

isomorphically

^{embedded in} the group of the

automorphisms

of the abelian group

P,

^o.

The

assumptions

^{of Cor. 1 can be}

weakened,

^{but some caution}

is

needed;

^for

example,

^we may remark that the finiteness of P ^was

only

^{used to ensure} both the finite

exponent

^{and the}

nilpotency

^{of P.}

The former is not

required

^{to define a structure of on a}

p-group, since the

ring Z/p-Z

^{can be}

^replaced by

^the

ring

^of

p-adic integers;

but in this case, if g, ^E I

gj OF,

^then

gj

may ^not be

a verbal function on P.

However,

^{a more} substantial obstacle to

generalizations

^{is met if one wants to omit the}

nilpotency assumption

for

P,

^{as the}

homomorphism g - )

is defined

only

if the infinite _pro- ducts involved converge in P. On the ^other

hand,

^{there is no}

difficulty

in

formulating

^{Cor. 1} for any finite

nilpotent

group such that p ^---1 mod d for every

prime p dividing

the order of P.

8. In this

paragraph

^{we shall}

explain

^the connection between the

preceeding

theorems and the results of M. Lazard

([3]).

THEOREM 3. Let C be a binomial

ring eontain2ng let g be

^a-rz

element

of

⁼ gl g2 ... gd the

factorization of

^{Theor 1. Then}

for

every ^cc C we have

We

only give

^an outline of the

proof,

whose details would

require

first to prove ^an

improvement

of Theor. 1.1 of

[1 ],

^{in order to take}

into account the

degrees

^{of the}

polynomials ^Pu,

^Ru

(see

par.

2)

ⁱⁿ

the

single

variables. One finds that the

Cfx}-mapping

of C into pc defined

by

is such that the

degree of gu

^{in x does not exceed}

L(~c) .

On the other

hand,

the conditions

(g,)0"

^{for i}

= 1, 2, ... , d implies

^{that each}

e. vanishes for x ⁼ a,

oe 2, == 1;

^since

clearly e

vanishes also for the

(d+1)-th

^{value one deduces}

~u = 0

^whenever

Then

(gicJ)-C

^{=1 as we wanted.}

For j = 2,

^{... , d the}

proof

is similar.

COROLLARY 2. Let P be a p-group whose

n-generators subgroups

have

nilpotency

^{class not}

exceeding

d. Then every verbal

f unction

^{in n}

variables on P can be

uniquely

^{written I where}

gj

^is

e-homogerceous of degree j for

every

integer

^c.

The

proof

^is

immediate, by

^{Theor. 3 and the usual}

homomorphism.

There is a connection between these statements and some results of M. Lazard

([3]);

he introduced the

concept

of « suite

tipique &#x3E;&#x3E;

^and

proved

^{that such a} sequence ^can be

uniquely represented

^{as an infinite}

product ([3],

^p.

143)

where t is a

parameter assuming non-negative integer

^values

and b~

lies in a suitable

extension,

say

E,

of the free group F.

Roughly speaking,

this is the factorization of our Theor. 3

for g

⁼

g(1 )

^{if we}

let d tend to

infinity,

^thus

Zldl

tend to the field of

rationals Q

^and

F* ⁼

FZcdJ to FQ, in

the role of E. In the

following ^example

^this

connection will appear ^more

precise:

^let

as in

[3],

theor. 2.4. Now let us

apply

^{Theor. 3 to} the word g ⁼ X Y ⁼

=~1~2...~. We have f or

every ^c. In

particular,

^{if we let c assume}

integer

^{values t and embed}

F* in E we obtain

By comparison

^{it is}

easily

^seen that gi -

bl,

g2 ---

b2,

..., gd ---

Likewise,

^{if .P is a} p-group of class ^not

exceeding

p -

1,

^{then our}

Cor. 2

applied

^to

g = X Y

^for

d = p -1 yields substantially

^Lem-

ma 4.5 of

[3]; by letting

we

assign

^{to P a} structure of

Lie-algebra

from which the

original group-multiplication

^{is obtained}

through

the Hausdorff formula. The

following (open) questions naturally

^arise:

Under the

assumptions

^{of Cor. 1}

(i.e.

without any bound for the class of

P)

^{let us still write}

g = XY, gl (x, y ) = x -E- y, (g(x, y ) ) 2 =

-

y~.

^Then

P,

^{+ is a}

loop (Theorem 2),

but what kind of structure is

(P, -~--, Q, ~) ~

^Does there exist a formula

enabling

^{one to} reconstruct the group

multiplication starting

^{with the two new}

operations

REFERENCES

[1] R. H. BRUCK: Completion of free groups by ^means of infinite product representation, ^Math. Zeit., ¹¹⁸ (1970), ^9-29.

[2] ^{M. HALL} jr.: ^The

Theory

of

Groups,

Mac Millan Co., ^New York, ^1959.

[3] ^M. LAZARD: Sur les groupes nilpotents ^{et les anneaux de} Lie, ^Ann. E.N.S.,

71 (1954), ^101-190.

[4] B. SCIMEMI: p-groups, diagonalizable automorphisms ^and

loops,

^{Rend. Sem.}

Mat. Padova, ⁴⁵ (1971), ^199-221.

Manoscritto pervenuto ^{in redazione il 9 novembre 1972.}